1994/12/28 by Amnon Rosenmann, Rosenmann, Amnon
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Graph theory and applications #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.GR
paper · pdf · doi:10.48550/arxiv.math/9412204
LaTex, 23 pages, no figures
arxiv created 1994/12/28 · openalex publication_date 1994/12/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given the Cayley graph of a finitely generated group G, with respect to a presentation Gα with n generators, the quotient of the rank of the fundamental group of subgraphs of the Cayley graph by the cardinality of the set of vertices of the subgraphs gives rise to the definition of the normalized cyclomatic quotient Ξ(Gα). The asymptotic behavior of this quotient is similar to the asymptotic behavior of the quotient of the cardinality of the boundary of the subgraph by the cardinality of the subgraph. Using Følner's criterion for amenability one gets that Ξ(Gα) vanishes for infinite groups if and only if they are amenable. When G is finite then Ξ(Gα)=1/|G|, where |G|'> is the cardinality of G, and when G is non-amenable then 1-n≤Ξ(Gα)≤ 0, with Ξ(Gα)=1-n if and only if G is free of rank n. Thus we see that on special cases Ξ(Gα) takes the values of the Euler characteristic of G. Most of the paper is concerned with formulae for the value of Ξ(Gα) with respect to that of subgroups and factor groups, and with respect to the decomposition of the group into direct product and free product. Some of the formulae and bounds we get for Ξ(Gα) are similar to those given for the spectral radius of symmetric random walks on the graph of Gα, but this is not always the case. In the last section of the paper we define and touch very briefly the balanced cyclomatic quotient, which is defined on concentric balls in the graph and is related to the growth of G.